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  • Graphing y =:
  • x^4-2*x^3+1
  • x^3/(x-2)^2
  • x^3+x^2-2
  • x^3/(x-1)^2
  • Integral of d{x}:
  • sin(x)*cos(x)^2 sin(x)*cos(x)^2

  • Identical expressions

  • sin(x)*cos(x)^ two
  • sinus of (x) multiply by co sinus of e of (x) squared
  • sinus of (x) multiply by co sinus of e of (x) to the power of two
  • sin(x)*cos(x)2
  • sinx*cosx2
  • sin(x)*cos(x)²
  • sin(x)*cos(x) to the power of 2
  • sin(x)cos(x)^2
  • sin(x)cos(x)2
  • sinxcosx2
  • sinxcosx^2

  • Similar expressions

  • sinx*cosx^2
Plotting a function graph/ sin(x)*cos(x)^2

Graphing y = sin(x)*cos(x)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                 2   
f(x) = sin(x)*cos (x)
f(x)=sin⁡(x)cos⁡2(x)f{\left(x \right)} = \sin{\left(x \right)} \cos^{2}{\left(x \right)} 
f = sin(x)*cos(x)^2
The graph of the function
02468-8-6-4-2-10101.0-1.0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
sin⁡(x)cos⁡2(x)=0\sin{\left(x \right)} \cos^{2}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=−51.8362786906154x_{2} = -51.8362786906154
x3=89.5353907744432x_{3} = 89.5353907744432
x4=73.8274274722061x_{4} = 73.8274274722061
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=−75.398223686155x_{6} = -75.398223686155
x7=50.2654824574367x_{7} = 50.2654824574367
x8=34.5575191894877x_{8} = 34.5575191894877
x9=64.4026493118058x_{9} = 64.4026493118058
x10=51.8362788934209x_{10} = 51.8362788934209
x11=42.4115007327518x_{11} = 42.4115007327518
x12=−23.561945003804x_{12} = -23.561945003804
x13=−70.6858349962623x_{13} = -70.6858349962623
x14=0x_{14} = 0
x15=−1.57079642505341x_{15} = -1.57079642505341
x16=−64.4026492408158x_{16} = -64.4026492408158
x17=43.9822971502571x_{17} = 43.9822971502571
x18=−43.9822971502571x_{18} = -43.9822971502571
x19=80.1106131546315x_{19} = 80.1106131546315
x20=−80.1106125824842x_{20} = -80.1106125824842
x21=67.5442422018325x_{21} = 67.5442422018325
x22=−7.85398150264842x_{22} = -7.85398150264842
x23=−89.5353907394375x_{23} = -89.5353907394375
x24=−65.9734457253857x_{24} = -65.9734457253857
x25=23.5619450555027x_{25} = 23.5619450555027
x26=−86.3937978155375x_{26} = -86.3937978155375
x27=56.5486677646163x_{27} = 56.5486677646163
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=−29.8451300981866x_{29} = -29.8451300981866
x30=−95.8185758682892x_{30} = -95.8185758682892
x31=87.9645943005142x_{31} = 87.9645943005142
x32=59.6902604182061x_{32} = 59.6902604182061
x33=70.6858345559153x_{33} = 70.6858345559153
x34=−81.6814089933346x_{34} = -81.6814089933346
x35=72.2566310325652x_{35} = 72.2566310325652
x36=1.57079648184495x_{36} = 1.57079648184495
x37=−9.42477796076938x_{37} = -9.42477796076938
x38=−72.2566310325652x_{38} = -72.2566310325652
x39=81.6814089933346x_{39} = 81.6814089933346
x40=−31.4159265358979x_{40} = -31.4159265358979
x41=−61.261056881309x_{41} = -61.261056881309
x42=9.42477796076938x_{42} = 9.42477796076938
x43=92.6769831301454x_{43} = 92.6769831301454
x44=−94.2477796076938x_{44} = -94.2477796076938
x45=36.1283160593477x_{45} = 36.1283160593477
x46=−42.4115006663339x_{46} = -42.4115006663339
x47=−50.2654824574367x_{47} = -50.2654824574367
x48=28.2743338823081x_{48} = 28.2743338823081
x49=100.530964914873x_{49} = 100.530964914873
x50=−53.4070751110265x_{50} = -53.4070751110265
x51=94.2477796076938x_{51} = 94.2477796076938
x52=21.9911485751286x_{52} = 21.9911485751286
x53=95.8185760508519x_{53} = 95.8185760508519
x54=−58.1194640027517x_{54} = -58.1194640027517
x55=7.85398164444075x_{55} = 7.85398164444075
x56=−15.707963267949x_{56} = -15.707963267949
x57=65.9734457253857x_{57} = 65.9734457253857
x58=−6.28318530717959x_{58} = -6.28318530717959
x59=−67.5442421609972x_{59} = -67.5442421609972
x60=−45.5530935824522x_{60} = -45.5530935824522
x61=−39.2699083096144x_{61} = -39.2699083096144
x62=78.5398163397448x_{62} = 78.5398163397448
x63=−36.128315423197x_{63} = -36.128315423197
x64=15.707963267949x_{64} = 15.707963267949
x65=37.6991118430775x_{65} = 37.6991118430775
x66=−97.3893722612836x_{66} = -97.3893722612836
x67=45.5530936288414x_{67} = 45.5530936288414
x68=−14.13716684381x_{68} = -14.13716684381
x69=6.28318530717959x_{69} = 6.28318530717959
x70=−28.2743338823081x_{70} = -28.2743338823081
x71=29.8451303144929x_{71} = 29.8451303144929
x72=14.1371670924752x_{72} = 14.1371670924752
x73=−17.278759737384x_{73} = -17.278759737384
x74=26.7035374084741x_{74} = 26.7035374084741
x75=7.85398173541774x_{75} = 7.85398173541774
x76=12.5663706143592x_{76} = 12.5663706143592
x77=86.3937978909611x_{77} = 86.3937978909611
x78=−83.2522054524035x_{78} = -83.2522054524035
x79=−20.4203520921076x_{79} = -20.4203520921076
x80=48.6946859820148x_{80} = 48.6946859820148
x81=−95.818575585294x_{81} = -95.818575585294
x82=4.71238883532779x_{82} = 4.71238883532779
x83=−37.6991118430775x_{83} = -37.6991118430775
x84=−92.6769836764771x_{84} = -92.6769836764771
x85=20.4203521537986x_{85} = 20.4203521537986
x86=−73.8274272804402x_{86} = -73.8274272804402
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)*cos(x)^2.
sin⁡(0)cos⁡2(0)\sin{\left(0 \right)} \cos^{2}{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−2sin⁡2(x)cos⁡(x)+cos⁡3(x)=0- 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos^{3}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=−2atan⁡(5−26)x_{3} = - 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x4=2atan⁡(5−26)x_{4} = 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x5=−2atan⁡(26+5)x_{5} = - 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
x6=2atan⁡(26+5)x_{6} = 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
The values of the extrema at the points:
 -pi     
(----, 0)
  2      

 pi    
(--, 0)
 2     

        /   _____________\       /      /   _____________\\    /      /   _____________\\ 
        |  /         ___ |      2|      |  /         ___ ||    |      |  /         ___ || 
(-2*atan\\/  5 - 2*\/ 6  /, -cos \2*atan\\/  5 - 2*\/ 6  //*sin\2*atan\\/  5 - 2*\/ 6  //)

       /   _____________\      /      /   _____________\\    /      /   _____________\\ 
       |  /         ___ |     2|      |  /         ___ ||    |      |  /         ___ || 
(2*atan\\/  5 - 2*\/ 6  /, cos \2*atan\\/  5 - 2*\/ 6  //*sin\2*atan\\/  5 - 2*\/ 6  //)

        /   _____________\       /      /   _____________\\    /      /   _____________\\ 
        |  /         ___ |      2|      |  /         ___ ||    |      |  /         ___ || 
(-2*atan\\/  5 + 2*\/ 6  /, -cos \2*atan\\/  5 + 2*\/ 6  //*sin\2*atan\\/  5 + 2*\/ 6  //)

       /   _____________\      /      /   _____________\\    /      /   _____________\\ 
       |  /         ___ |     2|      |  /         ___ ||    |      |  /         ___ || 
(2*atan\\/  5 + 2*\/ 6  /, cos \2*atan\\/  5 + 2*\/ 6  //*sin\2*atan\\/  5 + 2*\/ 6  //)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=π2x_{1} = \frac{\pi}{2}
x2=−2atan⁡(5−26)x_{2} = - 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x3=−2atan⁡(26+5)x_{3} = - 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
Maxima of the function at points:
x3=−π2x_{3} = - \frac{\pi}{2}
x3=2atan⁡(5−26)x_{3} = 2 \operatorname{atan}{\left(\sqrt{5 - 2 \sqrt{6}} \right)}
x3=2atan⁡(26+5)x_{3} = 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−2atan⁡(26+5)]\left(-\infty, - 2 \operatorname{atan}{\left(\sqrt{2 \sqrt{6} + 5} \right)}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
(2sin⁡2(x)−7cos⁡2(x))sin⁡(x)=0\left(2 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−2atan⁡(711−627)x_{2} = - 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}
x3=2atan⁡(711−627)x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}
x4=−2atan⁡(762+117)x_{4} = - 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{6 \sqrt{2} + 11}}{7} \right)}
x5=2atan⁡(762+117)x_{5} = 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{6 \sqrt{2} + 11}}{7} \right)}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[2atan⁡(711−627),∞)\left[2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(711−627)]\left(-\infty, - 2 \operatorname{atan}{\left(\frac{\sqrt{7} \sqrt{11 - 6 \sqrt{2}}}{7} \right)}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)cos⁡2(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} \cos^{2}{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(x)cos⁡2(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} \cos^{2}{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)*cos(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
sin⁡(x)cos⁡2(x)=−sin⁡(x)cos⁡2(x)\sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{2}{\left(x \right)}
- No
sin⁡(x)cos⁡2(x)=sin⁡(x)cos⁡2(x)\sin{\left(x \right)} \cos^{2}{\left(x \right)} = \sin{\left(x \right)} \cos^{2}{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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